In the branch of mathematics called functional analysis, a complemented subspace of a topological vector space
Formally, topological direct sums strengthen the algebraic direct sum by requiring certain maps be continuous; the result retains many nice properties from the operation of direct sum in finite-dimensional vector spaces.
In general, classifying all complemented subspaces is a difficult problem, which has been solved only for some well-known Banach spaces.
is a morphism in the category of vector spaces — that is to say, linear.
In the category of vector spaces, finite products and coproducts coincide: algebraically,
, because the projection maps defined above act as inverses to the natural inclusion of
Then one can solve the problem in the vector subspaces and recombine to form an element of
In the category of topological vector spaces, that algebraic decomposition becomes less useful.
The definition of a topological vector space requires the addition map
[1] The categorical definition of direct sum, however, requires
if (and only if) any of the following equivalent conditions hold: The topological direct sum is also written
; whether the sum is in the topological or algebraic sense is usually clarified through context.
can matter quite strongly: every complemented vector subspace
is Banach, then an equivalent condition is For any two topological vector spaces
has the indiscrete topology, and so the algebraic projection is continuous.
a free topological vector subspace: for some set
Not all finite-codimensional vector subspaces of a TVS are closed, but those that are, do have complements.
This property characterizes Hilbert spaces within the class of Banach spaces: every infinite dimensional, non-Hilbert Banach space contains a closed uncomplemented subspace, a deep theorem of Joram Lindenstrauss and Lior Tzafriri.
Then the following are equivalent:[10] A complemented (vector) subspace of a Hausdorff space
[1][proof 2] From the existence of Hamel bases, every infinite-dimensional Banach space contains unclosed linear subspaces.
is a continuous linear surjection, then the following conditions are equivalent: (Note: This claim is an erroneous exercise given by Trèves.
as a topological complement, but we have just shown that no continuous right inverse can exist.
is also open (and thus a TVS homomorphism) then the claimed result holds.)
Topological vector spaces admit the following Cantor-Schröder-Bernstein–type theorem: The "self-splitting" assumptions that
cannot be removed: Tim Gowers showed in 1996 that there exist non-isomorphic Banach spaces
[12] Understanding the complemented subspaces of an arbitrary Banach space
up to isomorphism is a classical problem that has motivated much work in basis theory, particularly the development of absolutely summing operators.
Such spaces are called prime (when their only infinite-dimensional complemented subspaces are isomorphic to the original).
in fact, they admit uncountably many non-isomorphic complemented subspaces.
[13] An infinite-dimensional Banach space is called indecomposable whenever its only complemented subspaces are either finite-dimensional or -codimensional.